Phase Stability and Electronic Properties of Hybrid Organic–Inorganic Perovskite Solid Solution (CH(NH2)2)x(CH3NH3)1–xPb(BryI1–y)3 as a Function of Composition

Compositional mixing provides the means to maintain the structural stability of a hybrid organic–inorganic perovskite for efficient and robust photovoltaic applications. Here we present a theoretical, first-principles study of the electronic and energetic properties of the solid solution (CH(NH2)2)x(CH3NH3)1–xPbBryI1–y, the mixing of two organic molecules with various orientations, formamidinium and methylammonium, and two halides, bromide and iodide. Our results show the variation in the band gap as a function of composition (x and y) provides several candidates that exceed the 27.5% Schockley–Queisser efficiency. The variation in the composition of hybrid perovskite shows specific regions where either the hexagonal or cubic phase dominates. We discuss the balance between the band gap and phase stability and indicate regions where the phase transition temperature between cubic and hexagonal phases is far from room temperature, indicating that these compositions are more robust at room temperature against phase transitions.


Orientation of organic molecule in solid solution supercells
To represent the different orientations potentially possible of the organic molecules (methylammonium or formamidinium), two representative sets of orientations were used as described in the Tables S1 and S2. For the bulk MAPI (and also FAPI) systems, we considered further molecular orientations, by considering 45 • (60 • ) rotational steps for the cubic (hexagonal) phases, and relaxing all these combinations. From these, we chose the two sets below for the solid solution based on these presenting the greatest range of results.    The crystal structure of the hybrid perovskites were cubic, as indicated in References noted in Table S3 and thus all the interaxial angles are at 90 • . Our lattices were generated using the replicated lattice parameters. [1][2][3][4] To generate the basis within the crystal, we chose to have the centre of the molecule as the origin of the crystal structure because the atomic positions of species in the molecule can be calculated easily by its symmetry. Besides the molecule, the position of the B and X sites is determined in the following table.   (a) (b) (c) Figure S1: Fully geometrically optimised structure of MAPI in (a) alongside with partially optimised (b) tetragonal and (c) cubic phases.

Properties in Different Phases
The three optimised cubic-like structures of MAPI in various phases are shown in Figure   S1. The orthorhombic structure has uneven lattice parameters, varying from 6.276 to 6.395 A. The tetragonal structure was set to 6.248Å in a and b and 6.516Å in c. Moreover, the cubic structure was set to 6.333Å in all lattice parameters. Regarding the interaxial angles, all the phases maintain the angles within only 1.5 • away from right angles.
When relaxing the tetragonal and cubic structures fully, they return to the orthorhombic phase, showing the orthorhombic phase is more preferable at zero temperature. We calculate the Gibbs free energies (Eqn. (1)) to form MAPI in the tetragonal phase and cubic phase, given that temperature is assumed at 300 K, are 0.009 eV and -0.002 eV respectively. The little difference in Gibbs energy difference suggests that the orthorhombic and cubic phases are energetically favourable to transition to one another. On the other hand, the tetragonal phase is comparably unfavourable.
The band gaps of orthorhombic, tetragonal and cubic are 1.58, 1.52 and 1.59 eV, respectively, in Figure Table S10. The orthorhombic and cubic structures share a similar direct band gap of 0.01 eV difference, which is at the same high-symmetry k-point. This suggests that the orthorhombic phase should be used for further calculations and the electronic band structure can be reduced as the orthorhombic phase is slightly more preferable than the cubic phase at zero temperature.
In the case of the tetragonal band structure in Figure S2b, the band gap is 1.52 eV, in agreement with Baikie et al., 5 which is 0.06 eV lower than the gaps in the orthorhombic and cubic phase. This is due to the strain effect of a 4% longer lattice parameter along the vertical axis than that of the cubic phase. This leads to the band splitting near the conduction bands of the tetragonal structure from the degenerate CBM in the cubic structure. 7 Another study by Park et al. 8 suggests that the tetragonal MAPI should exhibit as high as 1.6 eV but the use of PbCl 2 indicates the composition should be mixed I-Cl with an unknown concentration of chlorine present that could lead to the increase in the band gap.
In summary, our calculations have shown that the band gaps vary by only 0.07 eV. The orthorhombic phase of MAPI exhibits the lowest band gap and its formation energy has shown to be the most favourable among the cuboid phases. For future investigation, the orthorhombic phase of hybrid perovskite will be taken into account.

Rotation Energy
The energy between different molecular orientations (rotational configurations) E rot is given by where E rot is the energy difference between the rotated configuration with respect to the aligned molecules. For simplicity, we term this as rotational energy. E tot is the total energy of the rotated-molecule structure, E aligned is the total energy of the system with aligned molecules for reference, and n is the number of unit cells studied.

Molecular Rotation of Metastable Frozen Structure
Here, we look into the effect of rotating a molecule within a geometrically optimised system that has aligned molecules. The rotation energies required to rotate a pre-aligned molecule in a geometrically optimised MAPI is shown in Figure S3a. The angle of rotation is with respect to the aligned orientation of the molecules. By comparing the rotation along the three axes, the energetic cost to rotate around φ and ψ are similar and higher than around θ due to the deformation of PbI 6 cage around the MA molecule. 10 The rotation energy required for the full rotational profile could go as high as 0.32 eV/(u.c.) in Figure S3a (0.15 eV/(u.c.) for FA molecule in Figure S4a) which will be highly unlikely, given that the

Metastable Rotational Configuration
Here, we investigate the potential metastable states of the bulk hybrid perovskites. To   For more accurate phonon frequencies and band structure, a larger supercell should have been used, but within our study, the computational cost would be unfeasible.

Validity of the use of functionals
In the main results, we showed that the use of the PBE functional and D3 vdW correction in DFT calculations resulted in an agreeable electronic band gap of hybrid perovskites to that observed in experimental studies. To further justify the adoption, we tested the structurally optimised structures of a single chemical unit of MAPI and FAPB using hybrid functional HSE06 or spin-orbit coupling (SOC) or both. These were Γ point only calculations, only probing the band structure at the centre of the Brillouin zone due to computational cost.  Variation of the band gap with changing ratio of methylammonium to formamidinium As discussed in the main text, there is little correlation between the ratio of methylammonium to formamidinium and either the volume, or the bandgap, as shown in Figure S10.
The reason for this lack of correlation is that when the system relaxes, the organic molecule compensates for the strain created by the choice of I or Br, and the local structure.

Crystal Coordinate of High Symmetry k-points
Band Gaps of Mixed Perovskites presented by phase Figure S13a shows the bandgaps of the two separate phases of hexagonal and cubic as a function of composition. We note that the lowest bandgaps (ignoring FAPI) are observed for the 50% mix of methylammonium and formamidinium and 100% iodine. However, our results indicate that this phase is ultimately unstable and will decay into the hexagonal phase.

Polyhedra tilt data
To further analyse the data used throughout this manuscript, we also examined the role of the tilting of the polyhedra in each of our supercells. We present these results in Figure S14a.
We find that the tilt is generally reduced with increasing formamidinium, but this does not correlate with the bandgap, which is more determined by the Br:I ratio as discussed in the main manuscript.